Label switching (i.e., the posterior distribution is invariant to switching component labels) is a problematic issue when using MCMC to estimate mixture models.
Is there a standard (as in widely accepted) methodology to deal with the issue?
If there is no standard approach then what are the pros and cons of the leading approaches to solve the label switching problem?
There is a nice and reasonably recent discussion of this problem here:
Christian P. Robert Multimodality and label switching: a
discussion. Workshop on mixtures, ICMS March 3, 2010.
Essentially, there are several standard strategies, and each has pros and cons. The most obvious thing to do is to formulate the prior in such a way as to ensure there is only one posterior mode (eg. order the means of the mixuture components), but this turns out to have a strange effect on the posterior, and therefore isn’t generally used. Next is to ignore the problem during sampling, and then post-process the output to re-label the components to keep the labels consistent. This is easy to implement and seems to work OK. The more sophisticated approaches re-label on-line, either by keeping a single mode, or deliberately randomly permuting the labels to ensure mixing over multiple modes. I quite like the latter approach, but this still leaves the problem of how to summarise the output meaningfully. However, I see that as a separate problem.